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Lines and Curves: A Practical Geometry Handbook

Victor Gutenmacher and N. B. Vasilyev
Publication Date: 
Number of Pages: 
[Reviewed by
Underwood Dudley
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This book was originally published in Russian in 1978 and translated into English by A. Kundu for a 1980 edition. The present edition is based on that translation.

It was written to be a geometry text for high school students enrolled in the Gelfand Correspondence School, but it is not an introduction: the authors assume that readers know basic plane and analytic geometry.

It consists of brief expository sections followed by problems that are non-trivial and will be new to most American readers. The first introductory problem (number 0.1) is to determine the path traced out by the midpoint of a ladder that, as ladders do, slides down a wall. There is a small probability that you know that it is a quarter-circle, but I doubt that you know what path is followed by the midpoint of a segment that joins the endpoints of the minute hands of two watches lying on a table (problem 1.28). Even if you guess that it is a circle, I expect you might have some difficulty proving it.

As those two problems illustrate, the authors often have objects in motion, so Geometer’s Sketchpad can come in handy. It would also be useful to have with you a good supply of mathematical and problem-solving talent. Though the authors assert that the material is accessible to advanced high-school students, they would have to be quite advanced. The first use for the book that came to my mind was for a college senior seminar or similar course. It can also be used as a source of unusual and interesting problems. Anyone whose interest in mathematics is not dead would want to try to find the solution to problem 3.15: to find, given a square ABCD, those points that are closer to AB than to BC, CD, and DA. There are hints, answers, and solutions for selected problems, but not for 3.15. There is no book like this one, and it is well worth buying.

The book is plentifully supplied with marginal diagrams but for some reason they are not referred to in the text, an annoyance. Perhaps Russian students were used to it and had no difficulty. But don’t let that turn you away from this fine book.

Underwood Dudley has retired from DePauw University and is now living in Florida.


Preface * Notation * Introduction * 1. Sets of Points * 2. The Alphabet * 3. Logical Combinations * 4. Maximum and Minimum * 5. Level Curves * 6. Quadratic Curves * 7. Rotations and Trajectories * 8. Drawings, Animation, and the Magic Triangle * Answers, Hints, Solutions * Summary of Results from Analytic Geometry * Some Facts from School Geometry * A Dozen Assignments * About Victor Gutenmacher * About N.B. Vasilyev